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A270681
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Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 190", based on the 5-celled von Neumann neighborhood.
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4
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1, 5, 12, 24, 32, 52, 60, 88, 96, 132, 140, 184, 192, 244, 252, 312, 320, 388, 396, 472, 480, 564, 572, 664, 672, 772, 780, 888, 896, 1012, 1020, 1144, 1152, 1284, 1292, 1432, 1440, 1588, 1596, 1752, 1760, 1924, 1932, 2104, 2112, 2292, 2300, 2488, 2496, 2692
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OFFSET
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0,2
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COMMENTS
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Initialized with a single black (ON) cell at stage zero.
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REFERENCES
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S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
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LINKS
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FORMULA
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a(n) = 3/2*(-1+(-1)^n)-(-5+(-1)^n)*n+n^2 for n>1.
a(n) = n^2+4*n for n>1 and even.
a(n) = n^2+6*n-3 for n>1 and odd.
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>4.
G.f.: (1+4*x+5*x^2+4*x^3-5*x^4-x^6) / ((1-x)^3*(1+x)^2).
(End)
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MATHEMATICA
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CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=190; stages=128;
rule=IntegerDigits[code, 2, 10];
g=2*stages+1; (* Maximum size of grid *)
a=PadLeft[{{1}}, {g, g}, 0, Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca=a;
ca=Table[ca=CAStep[rule, ca], {n, 1, stages+1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k=(Length[ca[[1]]]+1)/2;
ca=Table[Table[Part[ca[[n]][[j]], Range[k+1-n, k-1+n]], {j, k+1-n, k-1+n}], {n, 1, k}];
Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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